Axiom:Rank Axioms (Matroid)/Definition 2

Definition

Let $\rho: \powerset S \to \Z$ be a mapping from the power set of $S$ into the integers

$\rho$ is said to satisfy the rank axioms if and only if:

$(\text R 1')$	$:$	$\ds \forall X \in \powerset S:$	$\ds 0 \le \map \rho X \le \size X $
$(\text R 2')$	$:$	$\ds \forall X, Y \in \powerset S:$	$\ds X \subseteq Y \implies \map \rho X \le \map \rho Y $
$(\text R 3')$	$:$	$\ds \forall X, Y \in \powerset S:$	$\ds \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y $

1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 2.$ Axiom Systems for a Matroid, Theorem $3$

\((\text R 1')\)	$:$	\(\ds \forall X \in \powerset S:\)	\(\ds 0 \le \map \rho X \le \size X \)
\((\text R 2')\)	$:$	\(\ds \forall X, Y \in \powerset S:\)	\(\ds X \subseteq Y \implies \map \rho X \le \map \rho Y \)
\((\text R 3')\)	$:$	\(\ds \forall X, Y \in \powerset S:\)	\(\ds \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y \)